2-1. Dependency Measures
library(reshape2) # melt function
library(ggplot2) # ggplot function
library(pcaPP) # Fast Kendall function
library(energy) # Distance Correlation
library(Hmisc) # Hoeffding's D measure
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
##
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:dplyr':
##
## src, summarize
## The following object is masked from 'package:SeuratObject':
##
## Key
## The following object is masked from 'package:Seurat':
##
## Key
## The following objects are masked from 'package:base':
##
## format.pval, units
library(zebu) # Normalized Mutual Information
# library(minerva) # Maximum Information Coefficient
library(XICOR) # Chatterjee's Coefficient
# library(dHSIC) # Hilbert Schmidt Independence Criterion
library(VineCopula) # Blomqvist's Beta
make_cormat <- function(dat_mat){
matrix_dat <- matrix(nrow = ncol(dat_mat), ncol = ncol(dat_mat))
cor_mat_list <- list()
basic_cor <- c("pearson", "spearman")
# find each of the correlation matrices with Pearson, Spearman, Kendall Correlation Coefficients
for (i in 1:2){
print(i)
cor_mat <- stats::cor(dat_mat, method = basic_cor[i])
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
}
# functions that take matrix or data.frame as input
no_loop_function <- c(pcaPP::cor.fk, Hmisc::hoeffd,
VineCopula::BetaMatrix)
for (i in 3:5){
print(i)
fun <- no_loop_function[[i-2]]
cor_mat <- fun(dat_mat)
if (i == 4){ # Hoeffding's D
cor_mat <- cor_mat$D
}
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
}
# functions that take two variables as input to calculate correlations.
need_loop <- c(zebu::lassie, energy::dcor2d, XICOR::calculateXI)
for (i in 6:8){
print(i)
fun <- need_loop[[i-5]]
cor_mat <- matrix(nrow = ncol(dat_mat),
ncol = ncol(dat_mat))
for (j in 2:ncol(dat_mat)){
for (k in 1:(j-1)){
if (i == 6){
cor_mat[j, k] <- fun(cbind(dat_mat[, j], dat_mat[, k]), continuous=c(1,2), breaks = 6, measure = "npmi")$global
} else {
cor_mat[j, k] <- fun(as.numeric(dat_mat[, j]),
as.numeric(dat_mat[, k]))
}
}
}
cor_mat[upper.tri(cor_mat, diag = T)] <- NA
cor_mat_list[[i]] <- cor_mat
}
return(cor_mat_list)
}
draw_heatmap <- function(cor_mat){
len <- 5
melted_cormat <- melt(cor_mat)
melted_cormat <- melted_cormat[!is.na(melted_cormat$value),]
break_vec <- round(as.numeric(quantile(melted_cormat$value,
probs = seq(0, 1, length.out = len),
na.rm = T)),
4)
break_vec[1] <- break_vec[1]-1
break_vec[len] <- break_vec[len]+1
melted_cormat$value <- cut(melted_cormat$value, breaks = break_vec)
heatmap_color <- unique(melted_cormat$value)
heatmap <- ggplot(data = melted_cormat, aes(x = Var2, y = Var1, fill = value))+
geom_tile(colour = "Black") +
ggplot2::scale_fill_manual(breaks = sort(heatmap_color),
values = rev(scales::viridis_pal(begin = 0, end = 1)
(length(heatmap_color)))) +
theme_bw() + # make the background white
theme(panel.border = element_blank(), panel.grid.major = element_blank(),
panel.grid.minor = element_blank(), axis.ticks = element_blank(),
# erase tick marks and labels
axis.text.x = element_blank(), axis.text.y = element_blank())
return (heatmap)
}
make_cor_heatmap <- function(dat_mat, cor_mat_list){
fun_lable <- c("Pearson's Correlation", "Spearman's Correlation", "Kendall's Correlation",
"Hoeffding's D", "Blomqvist's Beta", "NMI",
"Distance Correlation", "XI Correlation")
cor_heatmap_list <- list()
cor_abs_heatmap_list <- list()
# make correlation matrices
#cor_mat_list <- make_cormat(dat_mat)
for (i in 1:8){
print(i)
if(i == 5) {
cor_heatmap_list[[i]] <- NULL
cor_abs_heatmap_list[[i]] <- NULL
next
}
cor_mat <- abs(cor_mat_list[[i]])
# get heatmaps
cor_heatmap <- draw_heatmap(cor_mat)
# ggplot labels
ggplot_labs <- labs(title = paste("Heatmap of", fun_lable[i]),
x = "",
y = "",
fill = "Coefficient") # change the title and legend label
cor_heatmap_list[[i]] <- cor_heatmap + ggplot_labs
if (i %in% c(1,2,3,4,6)){
cor_abs_mat <- abs(cor_mat_list[[i]])
cor_abs_heatmap <- draw_heatmap(cor_abs_mat)
ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
x = "", # change the title and legend label
y = "",
fill = "Coefficient")
cor_abs_heatmap_list[[i]] <- cor_abs_heatmap + ggplot_abs_labs
} else {
ggplot_abs_labs <- labs(title = paste("Abs Heatmap of", fun_lable[i]),
subtitle = "Equivalent to Non-Abs Heatmap",
x = "", # change the title and legend label
y = "",
fill = "Coefficient")
cor_abs_heatmap_list[[i]] <- cor_heatmap + ggplot_abs_labs
}
}
ans <- list(cor_heatmap_list, cor_abs_heatmap_list)
return (ans)
}
load("CD4_seurat_corr2.RData")
# save(cormat_list, heatmap_list, saver_mat, store_date,
# file = "CD4_seurat_corr2.RData")
# cormat_list <- make_cormat(saver_mat)
# heatmap_list <- make_cor_heatmap(saver_mat, cormat_list)
cor_pearson_mat <- cormat_list[[1]]; cor_spearman_mat <- cormat_list[[2]];
cor_kendall_mat <- cormat_list[[3]]; cor_hoeffd_mat <- cormat_list[[4]];
cor_blomqvist_mat <- cormat_list[[5]]; cor_MI_mat <- cormat_list[[6]];
cor_dist_mat <- cormat_list[[7]]; cor_XI_mat <- cormat_list[[8]];
1. Pearson’s correlation coefficient
- Pearson’s correlation is to measure linear dependency of data, X and Y
- \(-1 \leq \rho_{Pearson}(X, Y) \leq 1\)
- \(\rho_{Pearson}(X, Y) = \frac{\sum(x_i-\bar{x})(y_i -\bar{y})}{\sum(x_i-\bar{x})^2(y_i -\bar{y})^2}\)
cor_pearson_mat[1:5,1:5]
## IGKC HBA2 HBB HBA1 IGHA1
## IGKC NA NA NA NA NA
## HBA2 0.03360766 NA NA NA NA
## HBB 0.02048294 0.96939713 NA NA NA
## HBA1 0.06589510 0.97633835 0.986972826 NA NA
## IGHA1 0.50242775 0.01596036 0.007303465 0.03546479 NA
quantile(cor_pearson_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.790171530 -0.006763938 -0.001996166 0.004827661 1.000000000
quantile(abs(cor_pearson_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 1.386344e-18 2.406502e-03 6.309352e-03 1.379830e-02 1.000000e+00
# plot the smallest correlations
cor_pearson_vec <- sort(abs(cor_pearson_mat), decreasing = T)
plot(cor_pearson_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_pearson_mat) == cor_pearson_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_pearson_mat) == rev(cor_pearson_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_pearson_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[1]]

2. Spearman’s correlation coefficient
- It captures the monotonic relationship between data, X and Y
- \(-1 \leq \rho_{Spearman}(X,Y) \leq 1\)
- \(\rho_{Spearman} = 1 - \frac{6\sum{d_i^2}}{n(n^2-1)}\) where \(d_i\) is the difference between the ranks of \(x_i\) and \(y_i\)
cor_spearman_mat[1:5,1:5]
## IGKC HBA2 HBB HBA1 IGHA1
## IGKC NA NA NA NA NA
## HBA2 0.3868128 NA NA NA NA
## HBB 0.4134363 0.9407810 NA NA NA
## HBA1 0.4033735 0.9474558 0.9286280 NA NA
## IGHA1 0.5538631 0.2751407 0.3012491 0.2677616 NA
quantile(cor_spearman_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.861926228 -0.001335261 0.020749572 0.221332870 1.000000000
quantile(abs(cor_spearman_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 0.000000000 0.007690792 0.032266296 0.231727174 1.000000000
# plot the smallest correlations
cor_spearman_vec <- sort(abs(cor_spearman_mat), decreasing = T)
plot(cor_spearman_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_spearman_mat) == cor_spearman_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_spearman_mat) == rev(cor_spearman_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_spearman_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[2]]

3. Kendall’s Tau
- It is an alternative method to Spearman’s correlations, identifying monotonic relationships.
- \(-1 \leq \rho_{Kendall}(X,Y) \leq 1\)
- \(\rho_{Kendall}(X,Y) = \frac{\#\;concordant\;pairs - \#\;discordant \;pairs}{0.5n(n-1)}\)
cor_kendall_mat[1:5,1:5]
## IGKC HBA2 HBB HBA1 IGHA1
## IGKC NA NA NA NA NA
## HBA2 0.2660973 NA NA NA NA
## HBB 0.2851550 0.7874437 NA NA NA
## HBA1 0.2778262 0.8024215 0.7666779 NA NA
## IGHA1 0.3911323 0.1864774 0.2049764 0.1812993 NA
quantile(cor_kendall_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.673362647 -0.001300303 0.019174624 0.202522688 1.000000000
quantile(abs(cor_kendall_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 0.000000000 0.007219801 0.030312714 0.210479749 1.000000000
# plot the smallest correlations
cor_kendall_vec <- sort(abs(cor_kendall_mat), decreasing = T)
plot(cor_kendall_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_kendall_mat) == cor_kendall_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_kendall_mat) == rev(cor_kendall_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_kendall_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[3]]

4. Hoeffding’s D
- It tests the independence of data by calculating the distance between the product of the marginal distributions under the null hypothesis and the empirical bi-variate distribution.
- \(-1 \leq D(X,Y) \leq 1\)
- \(D(X,Y) = \frac{(n-2)(n-3)D_1+D_2-2(n-2)D_3}{n(n-1)(n-2)(n-3)(n-4)}\)
- \(D_1 = \sum_{i=1}^{n} Q_i(Q_i-1)\)
- \(D_2 = \sum_{i=1}^{n} (R_i-1)(R_i-2)(S_j-1)(S_j-2)\)
- \(D_3 = \sum_{i=1}^{n} (R_i-2)(S_i-2)Q_i\)
cor_hoeffd_mat[1:5,1:5]
## IGKC HBA2 HBB HBA1 IGHA1
## IGKC NA NA NA NA NA
## HBA2 0.04547920 NA NA NA NA
## HBB 0.05205833 0.52231121 NA NA NA
## HBA1 0.04956955 0.54588797 0.48581086 NA NA
## IGHA1 0.10250330 0.02191421 0.02636821 0.02063613 NA
quantile(cor_hoeffd_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.0004169447 -0.0003534625 -0.0002814756 0.0031924664 0.5588614687
# plot the smallest correlations
cor_hoeffd_vec <- sort(abs(cor_hoeffd_mat), decreasing = T)
plot(cor_hoeffd_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_hoeffd_mat) == cor_hoeffd_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_hoeffd_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_hoeffd_mat) == rev(cor_hoeffd_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_hoeffd_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[4]]

5. Blomqvist’s Beta
- It measures dependency between variables by constructing a two-way contingency table with the medians of each margin as cutting points.
- \(0 \leq \beta \leq 1\)
- \(\beta_n = \frac{n_1-n_2}{n_1+n_2} = \frac{2n_1}{n_1+n_2} - 1\)
- \(\beta = P\{(X-\tilde{x})(Y-\tilde{y})>0\} - P\{(X-\tilde{x})(Y-\tilde{y}) < 0\}\)
cor_blomqvist_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] -0.5688889 NA NA NA NA
## [3,] 0.8671111 -0.4600000 NA NA NA
## [4,] -0.9071111 0.6600000 -0.7982222 NA NA
## [5,] -0.9182222 0.5857778 -0.8111111 0.8844444 NA
quantile(cor_blomqvist_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -1 1 1 1 1
quantile(abs(cor_blomqvist_mat), na.rm = T)
## 0% 25% 50% 75% 100%
## 0.001333333 1.000000000 1.000000000 1.000000000 1.000000000
# plot the smallest correlations
cor_blomqvist_vec <- sort(abs(cor_blomqvist_mat), decreasing = T)
plot(cor_blomqvist_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_blomqvist_mat) == cor_blomqvist_vec[i], arr.ind = T)
idx1 <- idx[i,1]; idx2 <- idx[i,2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_blomqvist_mat) == rev(cor_blomqvist_vec)[i], arr.ind = T)
idx1 <- idx[1,1]; idx2 <- idx[1,2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_blomqvist_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[5]]
## NULL
6. Distance Correlation
- it is measure to identify non-linear relationships between two random variables with energy distances.
- distance correlation is calculated by dividing the distance covariance between X and Y by the product of their distance standard deviations.
- \(0 \leq dCor \leq 1\)
- \(dCor(X,Y) = \frac{dCov(Y,Y)}{\sqrt{dVar(X)dVar(Y)}}\)
- \(dCov(X, Y) = \sqrt{\frac{1}{n^2} \sum_{k=1, l=1}^{n} A_{k,l}B_{k,l}}\)
- \(dVar(X) = dCov(X,X) and dVar(Y) = dCov(Y, Y)\)
cor_dist_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.1153987 NA NA NA NA
## [3,] 0.1183309 0.84974651 NA NA NA
## [4,] 0.1342292 0.86772216 0.80209622 NA NA
## [5,] 0.2526241 0.05499639 0.05971892 0.0548047 NA
quantile(cor_dist_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## 0.000000e+00 7.085565e-05 4.321211e-04 5.439166e-03 1.000000e+00
# plot the smallest correlations
cor_dist_vec <- sort(abs(cor_dist_mat), decreasing = T)
plot(cor_dist_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_dist_mat) == cor_dist_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_dist_mat) == rev(cor_dist_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_dist_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[6]]

7. Normalized Mutual Information
- It measures how much one random variable gives information about the other. For example, High mutual information indicates a large reduction in uncertainty.
- \(0 \leq MI(X,Y) \leq 1\), as it is normalized.
- $MI(X,Y) = f_{X,Y} (x,y) log_2 ; dxdy $
- \(MI(X,Y) = \sum \sum p_{X,Y} (x,y) log \frac{p_{X,Y} (x,y)}{P_X(x)P_Y(y)}\)
cor_MI_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.0440143 NA NA NA NA
## [3,] 0.0440143 1.00000000 NA NA NA
## [4,] 0.0440143 1.00000000 1.00000000 NA NA
## [5,] 0.2136502 0.05290446 0.05290446 0.05290446 NA
quantile(cor_MI_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -1.119380e-18 9.092741e-03 2.832558e-02 7.751096e-02 1.000000e+00
# plot the smallest correlations
cor_MI_vec <- sort(abs(cor_MI_mat), decreasing = T)
plot(cor_MI_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_MI_mat) == cor_MI_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_MI_mat) == rev(cor_MI_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_MI_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[7]]

8. Chatterjee’s XI Correlation
- It measures the degree of dependence between the variables with concept of rank.
- \(0 \leq \xi_n \leq 1\)
- \(\xi_n(X,Y) = 1 - \frac{3\sum_{i=1}^{n-1} |r_{i+1} -r_i|}{n^2-1}\)
- \(\xi_n(X,Y) = 1 - \frac{n\sum_{i=1}^{n-1}|r_{i+1}-r_i}{2\sum_{i=1}^{n}l_i(n-l_i)}\)
cor_XI_mat[1:5,1:5]
## [,1] [,2] [,3] [,4] [,5]
## [1,] NA NA NA NA NA
## [2,] 0.1009619 NA NA NA NA
## [3,] 0.1125111 0.68139610 NA NA NA
## [4,] 0.1014173 0.70251135 0.65458743 NA NA
## [5,] 0.2087703 0.03512883 0.05167311 0.03598037 NA
quantile(cor_XI_mat, na.rm = T)
## 0% 25% 50% 75% 100%
## -0.0415879994 -0.0003705076 0.0218764219 0.2681462148 0.9966268892
# plot the smallest correlations
cor_XI_vec <- sort(abs(cor_XI_mat), decreasing = T)
plot(cor_XI_vec)

#plot the high correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_XI_mat) == cor_XI_vec[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}

#plot the lowest correlations
par(mfrow = c(2,2))
for(i in 1:4){
idx <- which(abs(cor_XI_mat) == rev(cor_XI_vec)[i], arr.ind = T)
idx1 <- idx[1]; idx2 <- idx[2]
plot(saver_mat[,idx1], saver_mat[,idx2], asp = T,
pch = 16, xlab = paste0(colnames(saver_mat)[idx1], ", (", idx1, ")"),
ylab = paste0(colnames(saver_mat)[idx2], ", (", idx2, ")"),
main = paste0("Correlation of ", round(cor_XI_mat[idx1, idx2], 3)))
}

Heatmap
heatmap_list[[1]][[8]]

quantile_mat <- c()
for (i in 1:length(cormat_list)){
quantile_mat <- rbind(quantile_mat,
quantile(abs(cormat_list[[i]]), probs = c(0.10, 0.95), na.rm = T))
}
rownames(quantile_mat) <- c("Pearson", "Spearman", "Kendall", "Hoeffding's D",
"Blomqvist's Beta","Dist. Corr", "NMI", "XI Corr")
quantile_mat
## 10% 95%
## Pearson 8.844389e-04 0.03502224
## Spearman 1.144231e-03 0.57049810
## Kendall 1.114172e-03 0.52502885
## Hoeffding's D 2.733574e-04 0.05633377
## Blomqvist's Beta 9.995556e-01 1.00000000
## Dist. Corr 2.513364e-03 0.29575625
## NMI 4.029217e-06 0.04152404
## XI Corr 3.705076e-04 0.73673423
# save(quantile_mat, store_date, file = "CD4_seurat_quantile2.RData")